Similar Polygons

Consider and prove each statement one at a time.

Proving the Conditional Statement

If two polygons are similar, then a similarity transformation that maps one polygon to the other exists. Consider how that relationship affects the corresponding angles and sides of the similar polygons.

• Angles: Similarity transformations preserve angle measures. That means the corresponding angles of the two polygons are congruent. ✓
• Sides: Similarity transformations map line segments to other line segments. The length of all line segments change according to the scale factor. That means the corresponding sides of the two polygons change proportionally. ✓

These observations conclude the proof of the conditional statement.

Proving the Converse

Consider two polygons with congruent corresponding angles and proportional corresponding sides. The proof here will be carried out for quadrilaterals ABCD and PQRS, but it can be generalized to any polygon.

Since the corresponding angles are congruent and the corresponding sides are proportional, the following statements are true. ∠ A&≅∠ P ∠ B&≅∠ Q ∠ C&≅∠ R ∠ D&≅∠ S PQ/AB=QR/BC&=RS/CD=SP/DA To show that the polygons ABCD and PQRS are similar, a similarity transformation can be built to map ABCD to PQRS. This can be done in several ways, so here is just an example of one possibility.

1

Translate One Vertex to the Corresponding Vertex

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Use a translation that moves A to P. The image of this translation of ABCD is A'B'C'D'.

2

Rotate to Get the Correct Direction of One Pair of Segments

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Use a rotation around the common vertex that moves B' to PQ. The image of this rotation of A'B'C'D' is A''B''C''D''.

The following table contains some observations about the position of points A'', B'', C'', and D'' relative to PQRS.

Observation	Justification
P=A''	The translation moves A to P and, since this is the center of rotation, it stays there.
B'' is on PQ	This is how the angle of rotation was chosen.
D'' is on PS	This is true, because by assumption ∠ A is congruent to ∠ P and because rigid motions preserve angle measures. Note that, in this case, the orientation of ABCD and PQRS is the same. If the orientations are different, then a reflection of A''B''C''D'' in line PQ is also needed to match the orientations of the polygons.

3

Dilate to Get a Second Vertex to the Corresponding Vertex

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Use a dilation from the common vertex to move B'' to Q. This dilation maps A''B''C''D'' to A'''B'''C'''D'''.

The following table contains some observations about the position of points A''', B''', C''', and D''' relative to PQRS.

Observation	Justification
P=A'''	The translation moves A to P and, since this is the center of rotation and also the dilation, it stays there.
Q=B'''	This is how the scale factor of the dilation was chosen.
S=D'''	Since translations and rotations are rigid motions, AB=A''B'' and AD=A''D''. It is assumed that PQ/AB=PS/AD, so the dilation that moves B'' to Q, also moves D'' to S.
C''' is on QR	It is assumed that ∠ B≅ ∠ Q. Since rigid motions and dilations preserve angles, this means that ∠ B'''≅ ∠ Q.
C''' is on SR	It is assumed that ∠ D≅ ∠ S. Since rigid motions and dilations preserve angles, this means that ∠ D'''≅ ∠ S.
R=C'''	Both R and C''' is the intersection of QR and SR.

The steps above give a similarity transformation that maps ABCD to PQRS, so these two quadrilaterals are similar. This proves the converse statement.